We presented for the first time the complete theory of the discrete transformations of the covariant Dirac field on the de Sitter manifold whose expanding and collapsing portions are models of universes. We focused on the group of discrete symmetries in the physical frames of the expanding universe which is of actual interest in cosmology pointing out that similar results on the collapsing portion can be obtained simply by substituting \(\omega _H\rightarrow -\omega _H\) . Moreover, we derived all the transformations generated by the discrete isometries, including the improper ones, using conformal frames where we have obvious technical advantages but obscure physical meaning as the improper isometries do not have a physical meaning being of a mere mathematical interest. Thus we may say that this paper complete the previous studies concerning the de Sitter continuous isometries [1]}, [2]}, [3]} generating the principal invariants of the Dirac theory on the de Sitter expanding universe [4]}, [5]}, [6]}.
